Let X1,,Xn be n independent observations of the number of trials necessary to observe the 5th...

 

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Let X1,,Xn be n independent observations of the number of trials necessary to observe the 5th success in Bernoulli trials for which the odds of success are 0, so that for each i = 1,. ,n, %3D (*:); e 5 1 1+0 1+0 I-5 P(X, = a): For this question, you must show the reasoning for each of your answers without reference to Mathematica. (a) Find the maximum likelihood estimator of 0. (b) What is the maximum likelihood estimate of 0 if n = 1, 11 = 5 ? (c) Find the Fisher information for 6. (d) What is the Cramér-Rao lower bound for the variance of unbiased estimators of O based on n observations. Let X1,,Xn be n independent observations of the number of trials necessary to observe the 5th success in Bernoulli trials for which the odds of success are 0, so that for each i = 1,. ,n, %3D (*:); e 5 1 1+0 1+0 I-5 P(X, = a): For this question, you must show the reasoning for each of your answers without reference to Mathematica. (a) Find the maximum likelihood estimator of 0. (b) What is the maximum likelihood estimate of 0 if n = 1, 11 = 5 ? (c) Find the Fisher information for 6. (d) What is the Cramér-Rao lower bound for the variance of unbiased estimators of O based on n observations.